On Mon, 12 Feb 2007 13:37:52 -0800, "Fred Marshall"
<fmarshallx@remove_the_x.acm.org> wrote:

>
  (snipped)
>
>Also, if you consider the roots in polar coordinates then the closer the 
>magnitude is to 1.0, the higher the Q of the resonance.
>
>And, the +/- angle of the complex root pair represents the frequency of the 
>resonance where fs=2*pi radians and z^-1 is a delay equivalent to 1/fs.
>
>Fred 

Hi Fred,
  you make a *VERY* good point.  Estimating the 
resonant frequencies by measuring the 
angle of the poles is only valid if the poles 
are quite close to the unit circle.

At one time I didn't realize that fact and 
I ended up causing myself trouble.

I wonder about mike7411's transfer function.
It sure seem like a homework problem to me, 
but only mike7411 knows for sure.

That transfer function is so strange in it's 
"form" I wonder what would be the educational 
benefit of studying such a transfer function?

See Ya,
[-Rick-]

Tim Wescott wrote:
> mike7 wrote:
> > I have a transfer function that is:
>
> > H(z) =         Kz^(-T)
> >        -----------------------
> >         1 - z^(-1) + K*z^(-T)
>
> > T is a time delay
> > K is a loop gain factor
>
> > I was wondering if there is a way to calculate what the resonant
> > frequency of this system is.  Any help is appreciated.
>
> > Thank you.

Does it strike anybody else as odd how we are increasedly getting very
similar questions?

:-)

>
> Presumably T is an integer -- if not, then your transfer function
> doesn't make much sense.

Why not? Fractional delays are (at least conceptually) just as valid
as, for example, fractional derivatives :-).

> If T is greater than three then it won't be
> 'the resonant frequency', it'll be 'the resonant frequencies' or 'the
> dominant resonance' (usually the lowest frequency one).
>
> Decide on your values of K and T.  Take the roots of the denominator
> polynomial.  

The OP is presumably looking for the roots as a function of K and T
(as was described in his other post). For general K and T, these are
hard to find.

> The pairs of roots with complex values are resonant.  You
> can find their frequencies and decide which one is dominant (probably
> the one that's closest to the unit circle, assuming the system is stable).


Regards,
Andor

"Tim Wescott" <tim@seemywebsite.com> wrote in message 
news:E-WdnaBBs_I_Sk3YnZ2dnUVZ_qOpnZ2d@web-ster.com...
> mike7411@gmail.com wrote:
>
>> I have a transfer function that is:
>>
>> H(z) =         Kz^(-T)
>>        -----------------------
>>         1 - z^(-1) + K*z^(-T)
>>
>> T is a time delay
>> K is a loop gain factor
>>
>> I was wondering if there is a way to calculate what the resonant
>> frequency of this system is.  Any help is appreciated.
>>
>> Thank you.
>>
> Presumably T is an integer -- if not, then your transfer function doesn't 
> make much sense.  If T is greater than three then it won't be 'the 
> resonant frequency', it'll be 'the resonant frequencies' or 'the dominant 
> resonance' (usually the lowest frequency one).
>
> Decide on your values of K and T.  Take the roots of the denominator 
> polynomial.  The pairs of roots with complex values are resonant.  You can 
> find their frequencies and decide which one is dominant (probably the one 
> that's closest to the unit circle, assuming the system is stable).
>
> -- 
>
> Tim Wescott
> Wescott Design Services
> http://www.wescottdesign.com
>

Also, if you consider the roots in polar coordinates then the closer the 
magnitude is to 1.0, the higher the Q of the resonance.

And, the +/- angle of the complex root pair represents the frequency of the 
resonance where fs=2*pi radians and z^-1 is a delay equivalent to 1/fs.

Fred 

mike7411@gmail.com wrote:

> I have a transfer function that is:
> 
> H(z) =         Kz^(-T)
>        -----------------------
>         1 - z^(-1) + K*z^(-T)
> 
> T is a time delay
> K is a loop gain factor
> 
> I was wondering if there is a way to calculate what the resonant
> frequency of this system is.  Any help is appreciated.
> 
> Thank you.
> 
Presumably T is an integer -- if not, then your transfer function 
doesn't make much sense.  If T is greater than three then it won't be 
'the resonant frequency', it'll be 'the resonant frequencies' or 'the 
dominant resonance' (usually the lowest frequency one).

Decide on your values of K and T.  Take the roots of the denominator 
polynomial.  The pairs of roots with complex values are resonant.  You 
can find their frequencies and decide which one is dominant (probably 
the one that's closest to the unit circle, assuming the system is stable).

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

I have a transfer function that is:

H(z) =         Kz^(-T)
       -----------------------
        1 - z^(-1) + K*z^(-T)

T is a time delay
K is a loop gain factor

I was wondering if there is a way to calculate what the resonant
frequency of this system is.  Any help is appreciated.

Thank you.